74 research outputs found
Long-time asymptotic analysis for defocusing Ablowitz-Ladik system with initial value in lower regularity
Recently, we have given the bijectivity for defocusing Ablowitz-Ladik
systems in the discrete Sobolev space by inverse spectral method.
Based on these results, the goal of this article is to investigate the
long-time asymptotic property for the initial-valued problem of the defocusing
Ablowitz-Ladik system with initial potential in lower regularity. The main idea
is to perform proper deformations and analysis to the corespondent
Riemann-Hilbert problem with the unit circle as the jump contour . As a
result, we show that when , the solution admits
Zakharov-Manakov type formula, and when , the solution
decays fast to zero
The Devil's Invention: Asymptotic, Superasymptotic and Hyperasymptotic Series
Singular perturbation methods, such as the method of multiple scales and the method of matched asymptotic expansions, give series in a small parameter ε which are asymptotic but (usually) divergent. In this survey, we use a plethora of examples to illustrate the cause of the divergence, and explain how this knowledge can be exploited to generate a 'hyperasymptotic' approximation. This adds a second asymptotic expansion, with different scaling assumptions about the size of various terms in the problem, to achieve a minimum error much smaller than the best possible with the original asymptotic series. (This rescale-and-add process can be repeated further.) Weakly nonlocal solitary waves are used as an illustration.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41670/1/10440_2004_Article_193995.pd
The N -soliton of the focusing nonlinear SchrÖdinger equation for N large
We present a detailed analysis of the solution of the focusing nonlinear SchrÖdinger equation with initial condition ψ( x , 0) = N sech( x ) in the limit N → ∞. We begin by presenting new and more accurate numerical reconstructions of the N -soliton by inverse scattering (numerical linear algebra) for N = 5, 10, 20, and 40. We then recast the inverse-scattering problem as a Riemann-Hilbert problem and provide a rigorous asymptotic analysis of this problem in the large- N limit. For those ( x, t ) where results have been obtained by other authors, we improve the error estimates from O ( N −1/3 ) to O ( N −1 ). We also analyze the Fourier power spectrum in this regime and relate the results to the optical phenomenon of supercontinuum generation. We then study the N -soliton for values of ( x, t ) where analysis has not been carried out before, and we uncover new phenomena. The main discovery of this paper is the mathematical mechanism for a secondary caustic (phase transition), which turns out to differ from the mechanism that generates the primary caustic. The mechanism for the generation of the secondary caustic depends essentially on the discrete nature of the spectrum of the N -soliton. Moreover, these results evidently cannot be recovered from an analysis of an ostensibly similar “condensed-pole” Riemann-Hilbert problem. © 2006 Wiley Periodicals, Inc.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/55992/1/20162_ftp.pd
Correlation Functions for a Chain of Short Range Oscillators
We consider a system of harmonic oscillators with short range interactions
and we study their correlation functions when the initial data is sampled with
respect to the Gibbs measure. Such correlation functions display rapid
oscillations that travel through the chain. We show that the correlation
functions always have two fastest peaks which move in opposite directions and
decay at rate for position and momentum correlations and as
for energy correlations. The shape of these peaks is
asymptotically described by the Airy function. Furthermore, the correlation
functions have some non generic peaks with lower decay rates. In particular,
there are peaks which decay at rate for position and
momentum correlators and with rate for energy correlators.
The shape of these peaks is described by the Pearcey integral. Crucial for our
analysis is an appropriate generalisation of spacings, i.e. differences of the
positions of neighbouring particles, that are used as spatial variables in the
case of nearest neighbour interactions. Using the theory of circulant matrices
we are able to introduce a quantity that retains both localisation and analytic
viability. This also allows us to define and analyse some additional quantities
used for nearest neighbour chains. Finally, we study numerically the evolution
of the correlation functions after adding nonlinear perturbations to our model.
Within the time range of our numerical simulations the asymptotic description
of the linear case seems to persist for small nonlinear perturbations while
stronger nonlinearities change shape and decay rates of the peaks
significantly.Comment: 25 pages, 6 figure
Correlation Functions for a Chain of Short Range Oscillators
We consider a system of harmonic oscillators with short range interactions and we study their correlation functions when the initial data is sampled with respect to the Gibbs measure. Such correlation functions display rapid oscillations that travel through the chain. We show that the correlation functions always have two fastest peaks which move in opposite directions and decay at rate t-13 for position and momentum correlations and as t-23 for energy correlations. The shape of these peaks is asymptotically described by the Airy function. Furthermore, the correlation functions have some non generic peaks with lower decay rates. In particular, there are peaks which decay at rate t-14 for position and momentum correlators and with rate t-12 for energy correlators. The shape of these peaks is described by the Pearcey integral. Crucial for our analysis is an appropriate generalisation of spacings, i.e. differences of the positions of neighbouring particles, that are used as spatial variables in the case of nearest neighbour interactions. Using the theory of circulant matrices we are able to introduce a quantity that retains both localisation and analytic viability. This also allows us to define and analyse some additional quantities used for nearest neighbour chains. Finally, we study numerically the evolution of the correlation functions after adding nonlinear perturbations to our model. Within the time range of our numerical simulations the asymptotic description of the linear case seems to persist for small nonlinear perturbations while stronger nonlinearities change shape and decay rates of the peaks significantly
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